Missing value imputation for predictive models

ABSTRACT

Provided are techniques for imputing a missing value for each of one or more predictor variables. Data is received from one or more data sources. For each of the one or more predictor variables, an imputation model is built based on information of a target variable; a type of imputation model to construct is determined based on the one or more data sources, a measurement level of the predictor variable, and a measurement level of the target variable; and the determined type of imputation model is constructed using basic statistics of the predictor variable and the target variable. The missing value is imputed for each of the one or more predictor variables using the data from the one or more data sources and one or more built imputation models to generate a completed data set.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims the benefit of U.S.patent application Ser. No. 13/403,863, filed Feb. 23, 2012, whichapplication is incorporated herein by reference in its entirety.

FIELD

Embodiments of the invention relate to missing value imputation forpredictive models.

BACKGROUND

Predictive models are widely used and are often built on demographic,survey, and other data that contain many missing values. When thesemissing values are not handled appropriately during model building, thenthe predictive model is not reliable, and any decision based on thepredictive model may result in losses for a company. In addition,existing techniques for imputation of missing data do not efficientlyhandle the very large and distributed data sources that are nowencountered in practice.

Some existing techniques for imputing missing values are: meanimputation, regression imputation, multiple imputation, an ExpectationMaximization (EM) algorithm, and a technique that imputes missing valueswhile building a predictive model.

Mean imputation replaces missing values of a continuous variable with amean value that is computed based on all non-missing records. Meanimputation may provide inaccurate results.

Regression imputation regresses the variable that has missing values onall other variables and then uses the regression equation to imputemissing values for that variable. Random errors can be added to theimputed values to overcome the problem in underestimating the variancein the imputed variable. Regression imputation may provide inaccurateresults if the data does not follow the assumptions of a linearregression model. For example, when the variable to be imputed is acategorical variable, the variable to be imputed has a non-linearrelationship with other variables used to impute. Moreover, regressionimputation may regress the variable with missing values on all othervariables, so the imputation model building is not based on some basicdescriptive statistics.

Multiple imputation builds imputation models for a variable that hasmissing values on other variables. The imputation model is a linear orlogistic regression model for a continuous or categorical variable thathas missing values, respectively. Multiple imputation imputes multiple,complete data sets by its imputation process. Then, an appropriatepredictive model is built on each complete data set, and the results ofthe multiple predictive models are combined. The iterative nature of theimputation process may require many data passes to impute a singlecomplete data set, and those data passes are multiplied when multiplecomplete data sets are created. Multiple imputation uses logisticregression for a categorical variable with missing values, and multipleimputation uses several data passes to obtain the solution for onevariable within one imputation because, unlike linear regression,logistic regression does not have a closed form solution and needs aniterative process in which each iteration means one data pass. Thus, theexistence of categorical variables with missing values would increasecomputation cost using multiple imputation.

An Expectation Maximization (EM) algorithm is an iterative techniquethat alternates between steps (1) and (2) until the process converges onstable estimates, where step (1) estimates the model parameters based onthe current data set, and step (2) imputes the missing values based onthose estimated parameters to update the data set. Then the fill-in dataset is used to re-estimate the parameters. Typically, the EM algorithmis used under a multivariate normal model and missing values are imputedbased on a regression model. Moreover, the EM algorithm is an iterativeprocess that requires many data passes. If the predictive model ofinterest is more complicated than a multivariate normal model, then theEM algorithm is a system of equations which has specific forms forspecific applications. Thus, applying the EM algorithm may require skillto obtain the custom-made solutions for different applications.

Another technique imputes missing values while building a predictivemodel. A population of solutions is created using the data set withmissing values, where each solution includes parameters of the model andthe missing values. Each of the solutions in a population is checked forfitness. After the fitness is checked, the solutions in a population aregenetically evolved to establish a successive population of solutions.The process of evolving and checking fitness is continued until astopping criterion is reached. This technique may need many runs ofpopulations of solutions to reach the stopping criterion.

SUMMARY

Provided are a computer implemented method, computer program product,and system for imputing a missing value for each of one or morepredictor variables. Data is received from one or more data sources. Foreach of the one or more predictor variables, an imputation model isbuilt based on information of a target variable; a type of imputationmodel to construct is determined based on the one or more data sources,a measurement level of the predictor variable, and a measurement levelof the target variable; and the determined type of imputation model isconstructed using basic statistics of the predictor variable and thetarget variable. The missing value is imputed for each of the one ormore predictor variables using the data from the one or more datasources and one or more built imputation models to generate a completeddata set.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the drawings, like reference numbers represent corresponding partsthroughout.

FIG. 1 illustrates, in a block diagram, a computing system in accordancewith certain embodiments.

FIG. 2 illustrates, in a flow diagram, missing value imputation forsmall data source in accordance with certain embodiments.

FIG. 3 illustrates, in a flow diagram, missing value imputation forlarge and distributed data sources in accordance with certainembodiments.

FIG. 4 illustrates, in a flow diagram, further details of missing valueimputation for large and distributed data sources in accordance withcertain embodiments.

FIG. 5 illustrates, in a flow diagram, processing to build one or moreimputation models in accordance with certain embodiments. FIG. 5 isformed by FIGS. 5A and 5B.

FIG. 6 illustrates, in a table, determination of a type of imputationmodel to construct based on data source type (small or large anddistributed) and whether predictor variable and target variablevariables are categorical or continuous in accordance with certainembodiments.

FIG. 7 illustrates, in a table, constructing an imputation model of thedetermined imputation model type using basic statistics between apredictor variable and a target variable determined using a single passover the data in accordance with certain embodiments.

FIG. 8 illustrates, in a graph, comparison of missing value imputationbetween one linear regression and two piecewise linear regressions inaccordance with certain embodiments.

FIG. 9 illustrates, in a graph, an example of how two values may be usedto replace a missing value in accordance with certain embodiments.

FIG. 10 illustrates, in a table, basic statistics in each combination ofthe predictor variable data bin and target variable category inaccordance with certain embodiments.

FIG. 11 depicts a cloud computing node in accordance with certainembodiments.

FIG. 12 depicts a cloud computing environment in accordance with certainembodiments.

FIG. 13 depicts abstraction model layers in accordance with certainembodiments.

DETAILED DESCRIPTION

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

Embodiments provide an efficient system of missing value imputation thathas the advantages of a two-step process ((1) impute missing valuesbased on the imputation models; and (2) use the complete data set forthe subsequent predictive modeling)) and the ability to model possiblenonlinear relationships by building piecewise linear regression modelsbetween the variable to be imputed (i.e., the predictor variable) andthe variable used to impute (i.e., the target variable). Embodiments useonly the target variable for the subsequent model building to imputemissing values in the predictor variables. Embodiments can handlemissing value imputation with reasonable accuracy for a small datasource or large and distributed data sources because embodiments requireonly one data pass to collect necessary statistics between the predictorvariables with missing values and the target variable to buildimputation models, which include piecewise linear regression models tobe built on a list of data bins that are arranged by the locations ofthe predictor variables. In embodiments, “m” data bins are created,records are sorted in the “m” data bins based on the values of thepredictor variables, and some basic statistics are gathered for buildinga linear model for each data bin.

FIG. 1 illustrates, in a block diagram, a computing system in accordancewith certain embodiments. In FIG. 1, a computing device 100 includes amissing value imputation system 110, which generates one or moreimputation models 120 and one or more imputation strategies 130. Thecomputing device 100 is coupled to a data store 150. The data store 150includes original data 160. The original data may be from a small datasource or may be from large and distributed data sources.

The missing value imputation system 110 provides an efficient system toimpute missing values of inputs/predictor variables for the subsequentmodel building processes on large and distributed data sources (e.g.,using a Map-Reduce approach). The Map-Reduce approach may be describedas a software framework that supports distributed computing on large anddistributed data sources.

First, for each predictor variable that has missing values, imputationmodels based only on the target variable are built independently ondifferent data sources and on different machines using the Mapfunctions. During this processing, validation samples are extractedrandomly across all data sources and merged into one global validationsample along with the collection of imputation models using the Reducefunction. Second, all imputation models are evaluated based on theglobal validation sample in a distributed manner using another set ofMap functions to select the top K models and form an ensemble model.Third, the ensemble model and a selected imputation strategy are sent toeach data source to impute missing values of the predictor variables.Note that missing values in the target variable would not be imputed,hence those records are not included in the subsequent model buildingprocesses. Finally, the complete data sets for all possible predictorvariables are used to build any models for prediction, discovery, andinterpretation of relationships between the target variable and a set ofthe predictor variables.

The missing value imputation system 110 chooses only the target variableto build imputation models because (1) if one variable should beselected to make imputation models building feasible, the targetvariable is the only variable related and relevant to all possiblepredictor variables with missing values; (2) imputation models for allpredictor variables with missing values can be built independently asthey only depend on the target variable, so it is not necessary to buildimputation models with all variables sequentially and iteratively whichis the process used by multiple imputation and many data passes can besaved; and (3) while some information may be lost by not using allpossible predictor variables to build the imputation models, experimentsindicate the accuracy of results on the subsequent model buildingprocesses by using the target variable only are similar to or evenbetter than some conventional techniques (e.g., multiple imputationusing all other variables).

In embodiments, the missing value imputation system 110 builds piecewiselinear regression imputation models to catch the possible nonlinearrelationship between a predictor variable with missing values and thetarget variable, if both of them are continuous. To build piecewiselinear regression imputation models, the data values of the predictorvariable are aggregated into a list of data bins. In each data bin, onlybasic statistics (such as count, means, variance, and covariance) arecollected between the predictor variable and target variable, while thedata values are discarded, then a regression model is built for eachdata bin. For other measurement levels of the predictor variable andtarget variable, the imputation models are built based on some basicstatistics as well. A measurement level may be described as continuousor categorical. Therefore, the missing value imputation system 110 usesa single data pass to build imputation models for multiple predictorvariables with missing values and fully utilizes distributedcomputational resources to handle large data. The missing valueimputation system 110 uses a limited amount of memory to build eachimputation model, and the missing value imputation system 110 uses thetarget variable to build imputation models so only some univariate andbivariate statistics for predictor variables with missing values and thetarget variable are needed.

FIG. 2 illustrates, in a flow diagram, missing value imputation for asmall data source in accordance with certain embodiments. In certainembodiments, a small data source is a single data source. Control beginsat block 200 with the missing value imputation system 110 receiving data202 from a data source. The data 202 is input to block 204 and block208. In block 204, the missing value imputation system 110 builds animputation model 206 based on target variable information. Theimputation model 206, in addition to the data 202, is input to block208. In block 208, the missing value imputation system 110 imputesmissing values and outputs completed data 210.

FIG. 3 illustrates, in a flow diagram, missing value imputation forlarge and distributed data sources in accordance with certainembodiments. Control begins at block 300 with the missing valueimputation system 110, for each of one or more predictor variables,building imputation models. In block 300, multiple data sources aredistributed into Mappers. A Mapper may be described as a module thatruns a computation operation on a data source to generate the requiredresults. An imputation model between each predictor variable withmissing values and a target variable is built in each Mapperindependently. The kind of imputation model that is built depends on themeasurement levels of the predictor variable and target variables. TheMappers will pass imputation models for all predictor variables to asingle Reducer. A Reducer may be described as a module that merges theintermediate results from different Mappers and outputs final results.The Reducer has a collection of N imputation models for each of the oneor more predictor variables with missing values.

In block 302, the missing value imputation system 110, for a combinationof the one or more predictor variables, extracts validation samplesrandomly across the multiple data sources and merges the extractedvalidation samples into a global validation sample. In particular, inblock 302, a validation sample is randomly extracted from each Mapperindependently and the Reducer merges the validation samples into aglobal validation sample. In certain embodiments, the combination may beall the predictor variables. Block 300 and block 302 represent the firstmap reduce job.

In block 304, the missing value imputation system 110, for each of theone or more predictor variables, evaluates the imputation models basedon the global validation sample and selecting a top number (i.e., a topK) of the imputation models to form an ensemble model. Block 304represents a second map reduce job. In block 304, a global validationsample is scored by each Mapper to evaluate the accuracy of animputation model. The Reducer selects the top K imputation models out ofN possible imputation models based on some accuracy measures as thefinal ensemble model for each of the one or more predictor variableswith missing values.

In block 306, the missing value imputation system 110 imputes themissing value for each of the one or more predictor variables using thedata from the multiple data sources, one or more formed ensemble models,and a selected imputation strategy from the one or more imputationstrategies 130. Block 306 represents a third map reduce job. In certainembodiments, the missing value imputation system 110 may be implementedin Hadoop with a Map-Reduce interface or other comparable systems inorder to handle large and distributed data sources and computation inparallel. Hadoop may be described as a software framework that supportsdata-intensive distributed applications and enables applications to workwith thousands of nodes and petabytes of data. In block 306, the finalensemble model for each of the one and more predictor variables withmissing values, and an imputation strategy are sent to all Mappers toimpute the missing value for each of the one or more predictorvariables. If the subsequent model building processes would also runcomputations in a distributed manner using the Map-Reduce, then there isno need to have a Reducer in block 306. However, if the complete dataset should be exported, then the Reducer is used to gather datatogether.

The imputation strategy may be defined according to how the completeddata set (for all possible predictor variables) would be generated. Onepossible strategy is to impute the missing value for each of the one ormore predictor variables by the mean of K predicted values for thecontinuous predictor variable from the final ensemble model and by themode of K predicted values for the categorical predictor variable.Another possible strategy is to impute the missing value for each of theone or more predictor variables by the predicted value from a randomlyselected imputation model out of K models for the predictor variable tobe imputed.

FIG. 4 illustrates, in a flow diagram, further details of missing valueimputation for large and distributed data sources in accordance withcertain embodiments. In FIG. 4, there are 3 map reduce jobs. Section 420represents the first map reduce job. Section 430 represents the secondmap reduce job. Section 440 represents the third map reduce job.

In Section 420, each Mapper 400, 402, 404 receives data from a datasource, builds an imputation model for each of the one or more predictorvariables to output the imputation model, and samples data to output avalidation sample. At the Reducer 406, the imputation models from theMappers 400, 402, 404 are collected to form the collected imputationmodels for each of the one or more predictor variables. Also, at thereducer 406, the validation samples from the Mappers 400, 402, 404 aremerged to form the global validation sample.

In Section 430, the global validation sample is the input to score animputation model and a scoring result is the output for each of the oneor more predictor variables and for each Mapper 410, 412, 414. TheReducer 416 receives the scoring results from each Mapper 410, 412, 414and form an ensemble model by selecting the top K imputation modelsbased on the scoring results for each of the one or more predictorvariables. Without loss of generality, the ensemble model is denoted asModel 1, . . . , Model K for each of the one or more predictorvariables.

In Selection 440, each Mapper 422, 424, 426 imputes the missing valuefor each of the one or more predictor variables based on a data source,one or more ensemble models and a selected imputation strategy to outputthe completed data.

Unlike regression imputation and multiple imputation methods, themissing value imputation system 110 uses only the target variable toimpute missing values in predictor variables. Thus only univariate andbivariate statistics between the target variable and a predictorvariable with missing values are used to build imputation models,regardless of their measurement levels, and those statistics can becomputed for all predictor variables within each Mapper or data sourceindependently.

To catch possible nonlinear relationships between a predictor variablewith missing values and the target variable, if both of them arecontinuous, the missing value imputation system 110 builds piecewiselinear regression models on a list of data bins of the predictorvariable for each Mapper.

Only a single data pass is needed to build the imputation models for allpredictor variables with missing values as they are built on each Mapperindependently. Such an approach runs computations in a distributedmanner and can handle large and distributed data sources efficiently.

Using an ensemble of imputation models instead of only one imputationmodel to impute missing values gives more robust results for anyimputation strategy because outliers would only affect a few imputationmodels and those models are unlikely to be selected into the finalensemble model.

FIG. 5 illustrates, in a flow diagram, processing to build one or moreimputation models in accordance with certain embodiments. FIG. 5 isformed by FIGS. 5A and 5B. According to the measurement levels of thepredictor variable X and the target variable Y, four types of imputationmodels for the predictor variable X may be built.

Control begins at block 500 with the missing value imputation system 110determining whether the measured level of the predictor variable X iscontinuous. If so, processing continues to block 502, otherwise,processing continues to block 512 (FIG. 5B).

In block 502, the missing value imputation system 110 determines whetherthe measured level of the target variable Y is continuous. If so,processing continues to block 504, otherwise, processing continues toblock 508.

In block 504, the missing value imputation system 110 sorts records intodata bins based on the predictor variable X values and collectsstatistics, for each of the data bins, including: (1) a number ofrecords, (2) a mean of the predictor variable X, (3) a mean of thetarget variable Y, (4) a variance of the target variable Y, and (5) acovariance of the predictor variable X and the target variable Y. Inparticular, the missing value imputation system 110 assigns records intodata bins, but does not store the records in the data bins. Instead, themissing value imputation system 110 saves statistics (based on therecords) and discards the records. In block 506, the missing valueimputation system 110 builds one or more piecewise linear regressionimputation models using the statistics collected in block 504.

If the predictor variable and target variable are both continuous, thenthe piecewise linear regression models of the predictor variable on thetarget variable are built as imputation models because they mightcapture a possible non-linear relationship between the predictorvariable and the target variable. FIG. 8 illustrates, in a graph 800,comparison of missing value imputation between one linear regression andtwo piecewise linear regressions in accordance with certain embodiments.Using FIG. 8 as an example, suppose the true relation between predictorvariable X and target variable Y is a parabola 810 (the solid line), buta linear regression 820 (the dotted line) is built to impute a missingvalue x_(k), then the imputed value based on the target variable valueof y_(k) is x_(k)″. On the other hand, if two piecewise linearregressions 830 (the dashed line) are built, then the imputed value isx_(k)′, which is more accurate than x_(k)″.

To build piecewise linear regression models as imputation models, thepredictor variable values, as well as, the corresponding target variablevalues are assigned into a list of data bins based on the locations ofthe predictor variable. The approach is to sort predictor variable wherethe data bins are usually used to compute order statistics.

Suppose that there are m bins (a₁,a₂], (a₂,a₃], . . . , (a_(m),a_(m+i)],where a₁=−∞ and a_(m+1)=∞ after collecting and merging data bins thenthe following basic statistics in the i^(th) data bin are collected:

-   -   The number of records: n^((i))    -   Mean of Y: Y ^((i)),    -   Mean of X: X ^((i)),    -   Variance of Y: S_(YY) ^((i)), and    -   Covariance of X and Y: S_(XY) ^((i)).

Then the m piecewise linear regression models are built as follows:

X=β ₀ ^((i))+β₁ ^((i)) Y

where β₁ ^((i))=S_(XY) ^((i))/S_(YY) ^((i)) and β₀ ^((i))X= X ^((i))−β₁^((i)) Y ^((i)), i=1, . . . m.

If the k^(th) record is a missing value in prediction X with a knowntarget variable category, y_(k) then piecewise linear regression modelsare used to impute the missing value and an appropriate data bin has tobe selected first. If there is only one data bin in which the conditionβ₀ ^((i))+β₁ ^((i))y_(k)ε(a_(i),a_(i+1)] holds, then the linearregression model in the i^(th) data bin is used. However, there mayexist more than one data bin that satisfies the above condition, say β₀^((i))+β₁ ^((i))y_(k)ε(a_(i),a_(i+1)] and β₀ ^((j))+β₁^((j))y_(k)ε(a_(j),a_(j+1)]

FIG. 9 illustrates, in a graph 900, an example of how two values may beused to replace a missing value in accordance with certain embodiments.Using FIG. 9 as an example, a known target variable value y_(k) isapplied to m models, then two scores x_(k)′ in the i^(th) data bin andx_(k)″ in the j^(th) data bin may be used to impute the missing value ofX. Under this circumstance, a random number from the Bernoullidistribution B(p), which is a special case of binomial distribution with1 trial, where p=n^((i))/(n^((i))+n^((i))), is generated to determinewhich score should be used. If the random number is 1, then the missingvalue is imputed by x_(k)′ in the i^(th) data bin, otherwise, x_(k)″ inthe j^(th) data bin is used. This technique may be generalized to thesituation of more than two data bins by using a categoricaldistribution, which is a special case of multinomial distribution with 1trial.

In block 508, the missing value imputation system 110 sorts records intodata bins based on the predictor variable X values and collectsstatistics, for each category of the target variable Y and each of thedata bins, including: (1) a number of records and (2) a mean of thepredictor variable X. As mentioned above, the missing value imputationsystem 110 assigns records into data bins, but does not store therecords in the data bins. Instead, the missing value imputation system110 saves statistics (based on the records) and discards the records. Inblock 510, the missing value imputation system 110 builds one or morerobust conditional mean imputation models using the statistics collectedin block 508.

If the predictor variable is continuous, but the target variable iscategorical, then the predictor variable values, as well as, thecorresponding target variable categories are assigned into a list ofdata bins based on the locations of the predictor variable.

Suppose that there are m bins (a₁,a₂], (a₂,a₃], . . . , (a_(m),a_(m+1)],where a₁=−∞ and a_(m+1)=∞ after collecting and merging data bins, thenthe following basic statistics of X for the j^(th) category of Y, j=1, .. . , J, in the i^(th) data bin are collected:

-   -   The number of records: n^((i,j)), and    -   Mean of X: X ^((i,j)).

FIG. 10 illustrates, in a table 1000, basic statistics in eachcombination of the predictor variable data bin and target variablecategory in accordance with certain embodiments.

The first and the last S data bins are discarded in the following meancomputation, where S is a specified constant that does not vary with thesize of the data. Then, for each target variable category, the rest ofthe data bins are merged and the corresponding mean of the predictorvariable is obtained as follows:

${\overset{\_}{X}}_{Y = j} = {\frac{1}{\sum\limits_{i = {S + 1}}^{m - S}\; n^{({i,j})}}{\sum\limits_{i = {S + 1}}^{m - S}\; {n^{({i,j})}{{\overset{\_}{X}}^{({i,j})}.}}}}$

The conditional mean computed in this way is more robust because the Ssmallest values and S largest values as potential outliers are excluded.This is referred to herein as robust conditional mean imputation model.

If the k^(th) record is a missing value in predictor variable X with aknown target variable category, y_(k), then the missing value is imputedby the robust conditional mean of X conditional on Y=y_(k).

In block 512, the missing value imputation system 110 determines whetherthe measured level of the target variable Y is continuous. If so,processing continues to block 514, otherwise, processing continues toblock 518.

In block 514, the missing value imputation system 110 collectsstatistics, for each category of the predictor variable X, including:(1) a mean of the target variable Y and (2) a variance of the targetvariable Y. In block 516, the missing value imputation system 110 buildsone or more minimum z-score category imputation models using thestatistics collected in block 514.

For a categorical predictor variable with a continuous target variable,the mean and variance of a target variable in each category of thepredictor variable are collected to represent the target variable'sdistribution in the corresponding category. Suppose a categoricalpredictor variable X has 1, . . . , J categories, then the followingbasic statistics of Y for the j^(th) category of X are collected:

-   -   Mean of Y: Y ^((j))    -   Variance of Y: s_(YY) ^((j))

If the k^(th) record is a missing value in predictor variable X with aknown target variable value, y_(k), then, the missing value will beimputed with a predictor variable category by judging which distributionthe target variable value y_(k) is more likely to belong to, that is themissing value will be imputed as follows:

$x_{k} = {\arg {\; \;}{\min\limits_{j}{\{ {{\frac{y_{k} - {\overset{\_}{Y}}^{(j)}}{\sqrt{s_{YY}^{(j)}}}},{j = 1},\ldots \mspace{14mu},J} \}.}}}$

This is referred to herein as a minimum z-score category imputationmodel.

In block 518, the missing value imputation system 110 collectsstatistics, for each category combination of the predictor variable Xand the target variable Y, including: a number of records. In block 520,the missing value imputation system 110 builds one or more conditionalmode imputation models using the statistics collected in block 518.

If both the predictor variable and the target variable are categorical,then a contingency table is generated. That is, the number of records ineach category combination of the predictor variable and target variableis collected to form the contingency table. If the k^(th) record is amissing value in predictor variable X with a known target variablecategory, y_(k), then the missing value is imputed by the mode of Xconditional on Y=y_(k). This is referred to herein as a conditional modeimputation model.

FIG. 6 illustrates, in a table 600, determination of a type ofimputation model to construct based on data source type (small or largeand distributed) and whether predictor variable and target variables arecategorical or continuous in accordance with certain embodiments. Incertain embodiments, one data source is treated as a data source type of“small”, while multiple data sources are treated as a data source typeof “large and distributed”.

FIG. 7 illustrates, in a table 700, constructing an imputation model ofthe determined imputation model type using basic statistics between apredictor variable and a target variable determined using a single passover the data in accordance with certain embodiments.

The missing value imputation system 110 builds imputation models basedonly on the target variable information and based on determining whichimputation model type to construct (which is based on the data sourcetype and the measurement levels of the predictor variable and targetvariable (i.e., categorical or continuous)).

The missing value imputation system 110 reduces the required resources(e.g., time) for imputing missing values and, thus, allows for missingvalue imputation to be used more frequently, especially for large orcomplex data sets.

The missing value imputation system 110 uses a single data pass nomatter the data source is small or large and distributed.

The missing value imputation system 110 uses one or more imputationmodels to generate one set of values for inputs with missing values.

The missing value imputation system 110 handles missing valueimputations first in order to build a reliable predictive model.

Thus, the missing value imputation system 110 builds one or moreimputation models to impute missing predictor variable values based onlyon the target variable information by determining the type of imputationmodel to construct based on the data source type and whether thepredictor variable and target variables are categorical or continuous.

It is understood in advance that although this disclosure includes adetailed description on cloud computing, implementation of the teachingsrecited herein are not limited to a cloud computing environment. Rather,embodiments of the present invention are capable of being implemented inconjunction with any other type of computing environment now known orlater developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g. networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client devices through athin client interface such as a web browser (e.g., web-based email). Theconsumer does not manage or control the underlying cloud infrastructureincluding network, servers, operating systems, storage, or evenindividual application capabilities, with the possible exception oflimited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting for loadbalancing between clouds).

A cloud computing environment is service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure comprising anetwork of interconnected nodes.

Referring now to FIG. 11, a schematic of an example of a cloud computingnode is shown. Cloud computing node 1110 is only one example of asuitable cloud computing node and is not intended to suggest anylimitation as to the scope of use or functionality of embodiments of theinvention described herein. Regardless, cloud computing node 1110 iscapable of being implemented and/or performing any of the functionalityset forth hereinabove.

In cloud computing node 1110 there is a computer system/server 1112,which is operational with numerous other general purpose or specialpurpose computing system environments or configurations. Examples ofwell-known computing systems, environments, and/or configurations thatmay be suitable for use with computer system/server 1112 include, butare not limited to, personal computer systems, server computer systems,thin clients, thick clients, handheld or laptop devices, multiprocessorsystems, microprocessor-based systems, set top boxes, programmableconsumer electronics, network PCs, minicomputer systems, mainframecomputer systems, and distributed cloud computing environments thatinclude any of the above systems or devices, and the like.

Computer system/server 1112 may be described in the general context ofcomputer system executable instructions, such as program modules, beingexecuted by a computer system. Generally, program modules may includeroutines, programs, objects, components, logic, data structures, and soon that perform particular tasks or implement particular abstract datatypes. Computer system/server 1112 may be practiced in distributed cloudcomputing environments where tasks are performed by remote processingdevices that are linked through a communications network. In adistributed cloud computing environment, program modules may be locatedin both local and remote computer system storage media including memorystorage devices.

As shown in FIG. 11, computer system/server 1112 in cloud computing node1110 is shown in the form of a general-purpose computing device. Thecomponents of computer system/server 1112 may include, but are notlimited to, one or more processors or processing units 1116, a systemmemory 1128, and a bus 1118 that couples various system componentsincluding system memory 1128 to processor 1116.

Bus 1118 represents one or more of any of several types of busstructures, including a memory bus or memory controller, a peripheralbus, an accelerated graphics port, and a processor or local bus usingany of a variety of bus architectures. By way of example, and notlimitation, such architectures include Industry Standard Architecture(ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA)bus, Video Electronics Standards Association (VESA) local bus, andPeripheral Component Interconnects (PCI) bus.

Computer system/server 1112 typically includes a variety of computersystem readable media. Such media may be any available media that isaccessible by computer system/server 1112, and it includes both volatileand non-volatile media, removable and non-removable media.

System memory 1128 can include computer system readable media in theform of volatile memory, such as random access memory (RAM) 1130 and/orcache memory 1132. Computer system/server 1112 may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 1134 can be provided forreading from and writing to a non-removable, non-volatile magnetic media(not shown and typically called a “hard drive”). Although not shown, amagnetic disk drive for reading from and writing to a removable,non-volatile magnetic disk (e.g., a “floppy disk”), and an optical diskdrive for reading from or writing to a removable, non-volatile opticaldisk such as a CD-ROM, DVD-ROM or other optical media can be provided.In such instances, each can be connected to bus 1118 by one or more datamedia interfaces. As will be further depicted and described below,memory 1128 may include at least one program product having a set (e.g.,at least one) of program modules that are configured to carry out thefunctions of embodiments of the invention.

Program/utility 1140, having a set (at least one) of program modules1142, may be stored in memory 1128 by way of example, and notlimitation, as well as an operating system, one or more applicationprograms, other program modules, and program data. Each of the operatingsystem, one or more application programs, other program modules, andprogram data or some combination thereof, may include an implementationof a networking environment. Program modules 1142 generally carry outthe functions and/or methodologies of embodiments of the invention asdescribed herein.

Computer system/server 1112 may also communicate with one or moreexternal devices 1114 such as a keyboard, a pointing device, a display1124, etc.; one or more devices that enable a user to interact withcomputer system/server 1112; and/or any devices (e.g., network card,modem, etc.) that enable computer system/server 1112 to communicate withone or more other computing devices. Such communication can occur viaInput/Output (I/O) interfaces 1122. Still yet, computer system/server1112 can communicate with one or more networks such as a local areanetwork (LAN), a general wide area network (WAN), and/or a publicnetwork (e.g., the Internet) via network adapter 1120. As depicted,network adapter 1120 communicates with the other components of computersystem/server 1112 via bus 1118. It should be understood that althoughnot shown, other hardware and/or software components could be used inconjunction with computer system/server 1112. Examples, include, but arenot limited to: microcode, device drivers, redundant processing units,external disk drive arrays, RAID systems, tape drives, and data archivalstorage systems, etc.

Referring now to FIG. 12, illustrative cloud computing environment 1250is depicted. As shown, cloud computing environment 1250 comprises one ormore cloud computing nodes 1110 with which local computing devices usedby cloud consumers, such as, for example, personal digital assistant(PDA) or cellular telephone 1254A, desktop computer 1254B, laptopcomputer 1254C, and/or automobile computer system 1254N may communicate.Nodes 1110 may communicate with one another. They may be grouped (notshown) physically or virtually, in one or more networks, such asPrivate, Community, Public, or Hybrid clouds as described hereinabove,or a combination thereof. This allows cloud computing environment 1250to offer infrastructure, platforms and/or software as services for whicha cloud consumer does not need to maintain resources on a localcomputing device. It is understood that the types of computing devices1254A-N shown in FIG. 12 are intended to be illustrative only and thatcomputing nodes 1110 and cloud computing environment 1250 cancommunicate with any type of computerized device over any type ofnetwork and/or network addressable connection (e.g., using a webbrowser).

Referring now to FIG. 13, a set of functional abstraction layersprovided by cloud computing environment 1250 (FIG. 12) is shown. Itshould be understood in advance that the components, layers, andfunctions shown in FIG. 13 are intended to be illustrative only andembodiments of the invention are not limited thereto. As depicted, thefollowing layers and corresponding functions are provided:

Hardware and software layer 1360 includes hardware and softwarecomponents. Examples of hardware components include mainframes, in oneexample IBM® zSeries® systems; RISC (Reduced Instruction Set Computer)architecture based servers, in one example IBM pSeries® systems; IBMxSeries® systems; IBM BladeCenter® systems; storage devices; networksand networking components. Examples of software components includenetwork application server software, in one example IBM WebSphere®application server software; and database software, in one example IBMDB2®, database software. (IBM, zSeries, pSeries, xSeries, BladeCenter,WebSphere, and DB2 are trademarks of International Business MachinesCorporation registered in many jurisdictions worldwide).

Virtualization layer 1362 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers;virtual storage; virtual networks, including virtual private networks;virtual applications and operating systems; and virtual clients.

In one example, management layer 1364 may provide the functionsdescribed below. Resource provisioning provides dynamic procurement ofcomputing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricingprovide cost tracking as resources are utilized within the cloudcomputing environment, and billing or invoicing for consumption of theseresources. In one example, these resources may comprise applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal provides access to the cloud computing environment forconsumers and system administrators. Service level management providescloud computing resource allocation and management such that requiredservice levels are met. Service Level Agreement (SLA) planning andfulfillment provide pre-arrangement for, and procurement of, cloudcomputing resources for which a future requirement is anticipated inaccordance with an SLA.

Workloads layer 1366 provides examples of functionality for which thecloud computing environment may be utilized. Examples of workloads andfunctions which may be provided from this layer include: mapping andnavigation; software development and lifecycle management; virtualclassroom education delivery; data analytics processing; transactionprocessing; and missing value imputation for predictive models.

Thus, in certain embodiments, software, implementing missing valueimputation for predictive models in accordance with embodimentsdescribed herein, is provided as a service in a cloud environment.

Additional Embodiment Details

As will be appreciated by one skilled in the art, aspects of the presentinvention may be embodied as a system, method or computer programproduct. Accordingly, aspects of the present invention may take the formof an entirely hardware embodiment, an entirely software embodiment(including firmware, resident software, micro-code, etc.) or anembodiment combining software and hardware aspects that may allgenerally be referred to herein as a “circuit,” “module” or “system.”Furthermore, aspects of the present invention may take the form of acomputer program product embodied in one or more computer readablemedium(s) having computer readable program code embodied thereon.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, or device, or any suitable combination of the foregoing. Morespecific examples (a non-exhaustive list) of the computer readablestorage medium would include the following: an electrical connectionhaving one or more wires, a portable computer diskette, a hard disk, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), an optical fiber,a portable compact disc read-only memory (CD-ROM), an optical storagedevice, a magnetic storage device, solid state memory, magnetic tape orany suitable combination of the foregoing. In the context of thisdocument, a computer readable storage medium may be any tangible mediumthat can contain, or store a program for use by or in connection with aninstruction execution system, apparatus, or device.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of thepresent invention may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Java, Smalltalk, C++ or the like and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may execute entirely on theuser's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network (LAN) or awide area network (WAN), or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider).

Aspects of the embodiments of the invention are described below withreference to flowchart illustrations and/or block diagrams of methods,apparatus (systems) and computer program products according toembodiments of the invention. It will be understood that each block ofthe flowchart illustrations and/or block diagrams, and combinations ofblocks in the flowchart illustrations and/or block diagrams, can beimplemented by computer program instructions. These computer programinstructions may be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, other programmable dataprocessing apparatus, or other devices to function in a particularmanner, such that the instructions stored in the computer readablemedium produce an article of manufacture including instructions whichimplement the function/act specified in the flowchart and/or blockdiagram block or blocks.

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational processing (e.g., operations or steps) to beperformed on the computer, other programmable apparatus or other devicesto produce a computer implemented process such that the instructionswhich execute on the computer or other programmable apparatus provideprocesses for implementing the functions/acts specified in the flowchartand/or block diagram block or blocks.

The code implementing the described operations may further beimplemented in hardware logic or circuitry (e.g., an integrated circuitchip, Programmable Gate Array (PGA), Application Specific IntegratedCircuit (ASIC), etc. The hardware logic may be coupled to a processor toperform operations.

Devices that are in communication with each other need not be incontinuous communication with each other, unless expressly specifiedotherwise. In addition, devices that are in communication with eachother may communicate directly or indirectly through one or moreintermediaries.

A description of an embodiment with several components in communicationwith each other does not imply that all such components are required. Onthe contrary a variety of optional components are described toillustrate the wide variety of possible embodiments of the presentinvention.

Further, although process steps, method steps, algorithms or the likemay be described in a sequential order, such processes, methods andalgorithms may be configured to work in alternate orders. In otherwords, any sequence or order of steps that may be described does notnecessarily indicate a requirement that the steps be performed in thatorder. The steps of processes described herein may be performed in anyorder practical. Further, some steps may be performed simultaneously.

When a single device or article is described herein, it will be readilyapparent that more than one device/article (whether or not theycooperate) may be used in place of a single device/article. Similarly,where more than one device or article is described herein (whether ornot they cooperate), it will be readily apparent that a singledevice/article may be used in place of the more than one device orarticle or a different number of devices/articles may be used instead ofthe shown number of devices or programs. The functionality and/or thefeatures of a device may be alternatively embodied by one or more otherdevices which are not explicitly described as having suchfunctionality/features. Thus, other embodiments of the present inventionneed not include the device itself.

The illustrated operations of flow diagrams show certain eventsoccurring in a certain order. In alternative embodiments, certainoperations may be performed in a different order, modified or removed.Moreover, operations may be added to the above described logic and stillconform to the described embodiments. Further, operations describedherein may occur sequentially or certain operations may be processed inparallel. Yet further, operations may be performed by a singleprocessing unit or by distributed processing units.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

The terms “an embodiment”, “embodiment”, “embodiments”, “theembodiment”, “the embodiments”, “one or more embodiments”, “someembodiments”, and “one embodiment” mean “one or more (but not all)embodiments of the present invention(s)” unless expressly specifiedotherwise.

The terms “including”, “comprising”, “having” and variations thereofmean “including but not limited to”, unless expressly specifiedotherwise.

The enumerated listing of items does not imply that any or all of theitems are mutually exclusive, unless expressly specified otherwise.

The corresponding structures, materials, acts, and equivalents of allmeans or step plus function elements in the claims below are intended toinclude any structure, material, or act for performing the function incombination with other claimed elements as specifically claimed. Thedescription of embodiments of the present invention has been presentedfor purposes of illustration and description, but is not intended to beexhaustive or limited to the invention in the form disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the invention.The embodiments were chosen and described in order to best explain theprinciples of the invention and the practical application, and to enableothers of ordinary skill in the art to understand the invention forvarious embodiments with various modifications as are suited to theparticular use contemplated.

The flowchart and block diagrams in the figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be executed substantiallyconcurrently, or the blocks may sometimes be executed in the reverseorder, depending upon the functionality involved. It will also be notedthat each block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration, can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

The foregoing description of embodiments of the invention has beenpresented for the purposes of illustration and description. It is notintended to be exhaustive or to limit the embodiments to the preciseform disclosed. Many modifications and variations are possible in lightof the above teaching. It is intended that the scope of the embodimentsbe limited not by this detailed description, but rather by the claimsappended hereto. The above specification, examples and data provide acomplete description of the manufacture and use of the composition ofthe embodiments. Since many embodiments may be made without departingfrom the spirit and scope of the invention, the embodiments reside inthe claims hereinafter appended or any subsequently-filed claims, andtheir equivalents.

1. A method for imputing a missing value for each of one or morepredictor variables, comprising: receiving data from one or more datasources; for each of the one or more predictor variables, performing:building an imputation model based on information of a target variable;determining a type of imputation model to construct based on the one ormore data sources, a measurement level of the predictor variable, and ameasurement level of the target variable; and constructing thedetermined type of imputation model using basic statistics of thepredictor variable and the target variable; and imputing the missingvalue for each of the one or more predictor variables using the datafrom the one or more data sources and one or more built imputationmodels to generate a completed data set.
 2. The method of claim 1,wherein the measurement level comprises one of continuous andcategorical.
 3. The method of claim 1, further comprising: determiningthat the predictor variable is continuous and the target variable iscontinuous; sorting records into data bins based on the predictorvariable values; collecting, for each of the data bins, statisticscomprising a number of records, a mean of the predictor variable, a meanof the target variable, a variance of the target variable, and acovariance of the predictor variable and the target variable;determining that the type of the imputation model is a piecewise linearregression imputation model; and building the piecewise linearregression imputation model using the collected statistics.
 4. Themethod of claim 1, further comprising: determining that the predictorvariable is continuous and the target variable is categorical; sortingrecords into data bins based on the predictor variable values;collecting, for each category of the target variable and each of thedata bins, statistics comprising a number of records and a mean of thepredictor variable; determining that the type of the imputation model isa robust conditional mean imputation model; and building the robustconditional mean imputation model using the collected statistics.
 5. Themethod of claim 1, further comprising: determining that the predictorvariable is categorical and the target variable is continuous;collecting, for each category of the predictor variable, statisticscomprising a mean of the target variable and a variance of the targetvariable; determining that the type of the imputation model is a minimumz-score category imputation model; and building the minimum z-scorecategory imputation model using the collected statistics.
 6. The methodof claim 1, further comprising: determining that the predictor variableis categorical and the target variable is categorical; collecting, foreach category combination of the predictor variable and the targetvariable, statistics comprising a number of records; determining thatthe type of the imputation model is a conditional mode imputation model;and building the conditional mode imputation model using the collectedstatistics.
 7. The method of claim 1, wherein the data is received frommultiple data sources, and further comprising: for each of the one ormore predictor variables, building an imputation model independently oneach data source; for a combination of the one or more predictorvariables, extracting validation samples randomly across the multipledata sources; and merging the extracted validation samples into a globalvalidation sample; for each of the one or more predictor variables,evaluating the imputation models based on the global validation sample;and selecting a top number of the imputation models to form an ensemblemodel; and imputing the missing value for each of the one or morepredictor variables using the data from the multiple data sources, oneor more formed ensemble models, and a selected imputation strategy. 8.The method of claim 1, further comprising: building the completed dataset for the one or more predictor variables; and using the completeddata set to build predictive models for prediction, discovery, andinterpretation of relationships between the target variable and the oneor more predictor variables.
 9. The method of claim 1, wherein softwareis provided as a service in a cloud environment.